The one-dimensional heat equation in the Alexiewicz norm

TL;DR

This paper studies the one-dimensional heat equation with initial data in the Alexiewicz norm, extending classical solutions to a broader class of distributions including derivatives of continuous functions.

Contribution

It establishes existence, uniqueness, and estimates of solutions for the heat equation with initial data in the space of distributions integrable via the continuous primitive integral, including derivatives of continuous functions.

Findings

Solution $u_t$ converges to initial data in Alexiewicz norm as $t o 0^+$

Boundedness estimate $ orm{f imes ext{heat kernel}}_ty \

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Abstract

A distribution on the real line has a continuous primitive integral if it is the distributional derivative of a function that is continuous on the extended real line. The space of distributions integrable in this sense is a Banach space that includes all functions integrable in the Lebesgue and Henstock--Kurzweil senses. The one-dimensional heat equation is considered with initial data that is integrable in the sense of the continuous primitive integral. Let $\Theta_t(x)=\exp(-x^2/(4t))/\sqrt{4\pi t}$ be the heat kernel. With initial data $f$ that is the distributional derivative of a continuous function, it is shown that $u_t(x):=u(x,t):=f\ast\Theta_t(x)$ is a classical solution of the heat equation $u_{11}=u_2$. The estimate $\|f\ast\Theta_t\|_\infty\leq\|f\|/\sqrt{\pi t}$ holds. The Alexiewicz norm is $\|f\|=\sup_I|\int_If|$, the supremum taken over all intervals. The initial data is taken on in the Alexiewicz norm, $\|u_t-f\|\to 0$ as $t\to 0^+$. The solution of the heat equation is unique under the assumptions that $\|u_t\|$ is bounded and $u_t\to f$ in the Alexiewicz norm for some integrable $f$. The heat equation is also considered with initial data that is the $n$th derivative of a continuous function and in weighted spaces such that $\int_{-\infty}^\infty f(x)\exp(-ax^2)\,dx$ exists for some $a>0$. Similar results are obtained.